From 1999-2018, 139 different species were observed. For these analyses, only species counted in at least three years were included, leaving 108 species for analysis of trends.
We are interested in estimating in the marginal rate of annual change in counts for species \(s\):
\[ \theta^s(x) = \frac{E[Y^{s}_i | X_i = x + 1]}{E[Y^{s}_i | X_i = x]}, \]
where \(Y^s_i\) denotes the observed count for a species for MBBS Orange County route \(i = 1, \dots, 11\) and \(X_i = 0, \dots, 19\) is the time (in years) since the start of the MBBS.
Let \(\mu^s(\beta) = log(E[Y_i|X_i = x]) = \beta_0 + \beta_1 x\). Under this model, \(\theta^s(x) = \beta_1\). We estimate \(\beta\) using the generalized estimating equation:
\[ \sum_{i = 1}^n \frac{\partial \mu}{\partial \beta} V_i^{-1} \{Y_i^s - \mu^s(\beta) \}, \]
where \(V^s_i\) is the so-called “working” correlation matrix.
For each species, parameters from each of the above models will be estimated. The model with the smallest mean absolute error (?) will be selected as the “best” model. Models where the fitter failed to converge will be excluded. The best model will be presented in the results table, but all model results will be available for review.